Since they need to last the night I would use the bubblers, at least at night when the water is still. I think the cories would probably be more comfortable all together since they’re schooling fish. When I moved mine I only separated for safety. Don’t want anything traveling with a roommate who might think it was yummy! So the rummies got their own bucket while the tiger barbs and white skirts all moved together. The whisker shrimp got their own bucket since I didn’t want them to go after the amanos in a tight space. Etc.
I would also highly recommend a black storage tote over buckets. Gives them more horizontal space and has a top to avoid too much splashing or anything falling in. Black (or anything opaque) to lower stress levels. I drilled a small hole, a little bigger than an airline tube in the top and then taped the bubbler to the top and ran the airline down into the water.
I didn’t heat the water during transit. Figured the risk of temp fluctuation back and forth was more likely to damage them then a day or two at room temperature. I slowly lowered the heater temp in the couple days prior to moving so they acclimated to room temp before the trip. At the new house, likewise, I increased the temp over a couple days to ease the transition back to their normal temp.
If you have long enough before the move I would get a few small sponge filters that you can start getting seeded, that way you can hook up a little sponge filter overnight in the tubs with the highest bioload (the larger fish). I also tossed the live filter media (rinsed well in aquarium water, so there’s not waste being added too)) into the tote with them along with a bit of purigen and one of those aqueon “pure” balls that’s supposed to help absorb/neutralize ammonia. I also put in a few drops of prime for that purpose.